[/quote]
,
.
be the feet of the altitudes from 
to 
respectively. Let 
be the circle with diameter 
.
be the intersection of 
and 
.
be the orthocenter of 
be the intersection of 
and 
be the lines through 
tangent to 
respectively. Let 
be the intersection of 
and 
.
and 
.
be the line through 
and let 
be the reflection of 
is tangent to 
.
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle 
with circumcenter 
is constant.
.
on 
and 
on 
analogously, using the excircles opposite 
and 
,
lies on the circumcircle of triangle 
is a function for which the expression 
for all 
is always a perfect square. Prove that 
for all 
.
.
and 
and 
are on the same side of 
be midpoint of segment 
be midpoint of segment 
with 
and 
that verify the following equation for all 
:
be positive real integer such that 
Prove that
such that 
and 
is a prime number
be a positive integer, and let its positive divisors be![\[
d_1 < d_2 < \cdots < d_k.
\]](http://latex.artofproblemsolving.com/6/3/e/63ec82aba0d6c4eece89af6a1247a89498616281.png)
Define 
to be the number of ordered pairs 
with 
such that 
.
.![\[
n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k},
\]](http://latex.artofproblemsolving.com/2/7/2/272dc2cc33ebbf695311cb61567fa0297e1299b6.png)
where the 
are distinct primes and the 
are positive integers.
.
meets the side 
and 
are 
and 
and 
and 
.
is a rectangle if and only if 
.
satisfying
Prove
and 
beautiful divisors and difference of any 
that verify the following equation for all 
:
Y by
Given a triangle 
we call, respectively, 
and 
the points of intersection of the bisector of 
with the circumscribed circle 
of and with the side 
. Let 
be a circle that passes through points 
and . We call 
and 
the points of intersection of with line 
and with circle respectively (
, 
). Prove that 
, and are collinear.
we call, respectively, 
and 
the points of intersection of the bisector of 
with the circumscribed circle 
of and with the side 
. Let 
be a circle that passes through points 
and . We call 
and 
the points of intersection of with line 
and with circle respectively (
, 
). Prove that 
, and are collinear.This post has been edited 2 times. Last edited by parmenides51, Dec 8, 2021, 6:43 PM
Reason: source
Reason: source
be the second intersection of 
and 
,
,
,
